Following the work of several authors, Hilbert's irreducibility theorem has been recast in geometric language as follows.
An irreducible polynomial \(p(t,x)\)  defines in a natural way an irreducible cover (that is, a finite surjective morphism) of an open subvariety of \(\mathbb {P}_1\)  over \(K\) : this is obtained simply by projecting the zero locus of \(p\)  on the \(t\) -line. Hilbert's irreducibility theorem can then be interpreted as stating that `many' fibres of this cover are irreducible over \(K\)  and hence, in particular, that they have no \(K\) -rational points; in turn, this is also equivalent to `many' rational points of \(\mathbb {P}_1\)  not being the image of rational points in the zero locus of \(p\) . Similar constructions can be made in the case of several variables, and this point of view is encapsulated in the following definition of Serre (for this and much more about the Hilbert property we refer the reader to the books of Serre [1]} and Fried and Jarden [2]}):
